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Abstract A direct way to spot structural features that are universally shared among proteins is to find analogues from simpler condensed matter systems. In the current study, the feasibility of creating ensembles of artificial structures that can automatically reproduce a large number of geometrical and topological descriptors of globular proteins is investigated. Towards this aim, a simple cubic (SC) arrangement is shown to provide the best background lattice after a careful analysis of the residue packing trends from 210 globular proteins. It is shown that a minimalistic set of rules imposed on this lattice is sufficient to generate structures that can mimic real proteins. In the proposed method, 210 such structures are generated by randomly removing residues (beads) from clusters that have a SC lattice arrangement such that all the generated structures have single connected components. Two additional sets are prepared from the initial structures via random relaxation and a reverse Monte Carlo simulated annealing algorithm, which targets the average radial distribution function (RDF) of 210 globular proteins. The initial and relaxed structures are compared to real proteins via RDF, bond orientational order parameters and several descriptors of network topology. Based on these features, results indicate that the structures generated with 40% occupancy closely resemble real residue networks. The structure generation mechanism automatically produces networks that are in the same topological class as globular proteins and reproduce small-world characteristics of high clustering and small shortest path lengths. Most notably, the established correspondence rules out icosahedral order as a relevant structural feature for residue networks in contrast to other amorphous systems where it is an inherent characteristic. The close correspondence is also observed in the vibrational characteristics as computed from the Anisotropic Network Model, therefore hinting at a non-superficial link between the proteins and the defect laden cubic crystalline order. 

Abstract The friendship paradox is the observation that the degrees of the neighbours of a node in any network will, on average, be greater than the degree of the node itself. In common parlance, your friends have more friends than you do. In this article, we develop the mathematical theory of the friendship paradox, both in general as well as for specific model networks, focusing not only on average behaviour but also on variation about the average and using generating function methods to calculate full distributions of quantities of interest. We compare the predictions of our theory with measurements on a large number of real-world network datasets and find remarkably good agreement. We also develop equivalent theory for the generalized friendship paradox, which compares characteristics of nodes other than degree to those of their neighbours. 

Abstract Preferential attachment (PA) models are a common class of graph models which have been used to explain why power-law distributions appear in the degree sequences of real network data. Among other properties of real-world networks, they commonly have non-trivial clustering coefficients due to an abundance of triangles as well as power laws in the eigenvalue spectra. Although there are triangle PA models and eigenvalue power laws in specific PA constructions, there are no results that existing constructions have both. In this article, we present a specific Triangle Generalized Preferential Attachment Model that, by construction, has non-trivial clustering. We further prove that this model has a power law in both the degree distribution and eigenvalue spectra.